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Chapter 5: Problem 41

Identify the slope and \(y\) -intercept, then graph the line. $$y=-\frac{3}{4} x+1$$

Short Answer

Expert verified
The slope of the given equation is \(-\frac{3}{4}\) and the y-intercept is \(1\). Plot the y-intercept point \((0, 1)\) on the graph. Then, use the slope to find another point on the line at \((4, -2)\). Draw a straight line through these points to graph the line \(y = -\frac{3}{4}x + 1\).

Step by step solution

01

Identify the Slope and Y-Intercept

We are given the equation \(y = -\frac{3}{4}x + 1\). Comparing it to the slope-intercept form, we can identify the slope and y-intercept as follows: - Slope (\(m\)): \(-\frac{3}{4}\) - Y-Intercept (\(b\)): \(1\)
02

Plot the Y-Intercept

Begin by plotting the y-intercept on the graph. Since the y-intercept is \(1\), this means the line crosses the y-axis at the point \((0, 1)\). Place a point on the graph at this location.
03

Plot Additional Points Using the Slope

To graph the line, we need to find more points on the line. We can use the slope to do this. The slope \(-\frac{3}{4}\) means that for every 4 units we move to the right on the x-axis, we move 3 units down on the y-axis. From the y-intercept point \((0, 1)\), we can move 4 units to the right and 3 units down to find another point on the line at \((4, -2)\). Place a point on the graph at this location.
04

Draw the Line

Now that we have at least two points on the graph (the y-intercept and the additional point found in Step 3), we can draw a straight line through these points. This line represents the graph of the given equation, \(y = -\frac{3}{4}x + 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form of a linear equation is a way to express linear relationships in a straightforward manner. It is given by the equation \( y = mx + b \), where:
  • \( y \) stands for the dependent variable (commonly the vertical axis on a graph),
  • \( x \) is the independent variable (commonly the horizontal axis),
  • \( m \) is the slope of the line, and
  • \( b \) is the y-intercept, or where the line crosses the y-axis.
To determine the slope and y-intercept from an equation in this form, simply match the components to the format of \( y = mx + b \). This makes graphing and interpreting the equation much easier.
Slope
The slope of a line indicates its steepness and direction. It is often represented by the letter \( m \) in the slope-intercept form of a line. Slope is calculated as "rise over run," which is the change in the vertical direction divided by the change in the horizontal direction between two points on a line.
  • A positive slope rises as it moves from left to right.
  • A negative slope falls as it moves from left to right.
  • A slope of zero indicates a horizontal line.
  • An undefined slope corresponds to a vertical line.
In the equation \( y = -\frac{3}{4}x + 1 \), the slope is \(-\frac{3}{4}\), meaning for each 4 units you move to the right, the line moves 3 units down.
Y-Intercept
The y-intercept of a line is where it crosses the y-axis. This point is represented as \( b \) in the slope-intercept form of an equation. It is found when the value of \( x \) is zero: essentially the starting point of the line when plotted on a graph.For example, in the equation \( y = -\frac{3}{4}x + 1 \), the y-intercept is \( 1 \). On a graph, you will find this point at (0, 1). It's crucial because it provides a definite starting place from which to plot the line using the slope.
Graphing Points
Graphing points involves plotting points on a coordinate plane based on their \((x, y)\) values. Once you have the key points, such as the y-intercept, you can use the slope to find additional points on the line.For the equation \( y = -\frac{3}{4}x + 1 \), you start by plotting the y-intercept at \((0, 1)\). To find another point, you apply the slope:
  • Starting at the y-intercept \((0, 1)\), move 4 units to the right (positive direction on the x-axis),
  • Then move 3 units down (negative direction on the y-axis) to reach the point \((4, -2)\).
Plot these points and draw a straight line through them, which gives the visual representation of the linear equation. This technique helps you visualize how the equation sets a relationship between the variables \( x \) and \( y \).

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Most popular questions from this chapter

Write a system of equations and solve. How many ounces of pure orange jurce and how many ounces of a citrus fruit drink containing \(5 \%\) fruit juice should be mixed to get 76 oz of a fruit drink that is \(25 \%\) fruit juice?

The measure of the smallest angle of a triangle is onethird the measure of the largest angle. The middle angle measures \(30^{\circ}\) less than the largest angle. Find the measures of the angles of the triangle. (Hint: Recall that the sum of the measures of the angles of a triangle is \(180^{\circ} .\) )

Write the slope-intercept form of the equation of the line, if possible, given the following information. $$m=-\frac{4}{9} \text { and } y \text { -intercept }(0,2)$$

Write an equation of the line perpendicular to the given line and containing the given point. Write the answer in slope intercept form or in standard form, as indicated. $$4 x-y=-3 ;(-8,-3) ; \text { standard form }$$

Solve each system \(\begin{aligned} 7 x+8 y-z &=16 \\\\-\frac{1}{2} x-2 y+\frac{3}{2} z &=1 \\\ \frac{4}{3} x+4 y-3 z &=-\frac{2}{3} \end{aligned}\)
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